Nonvanishing of derivatives of functions
Author:
Qingfeng Sun
Journal:
Proc. Amer. Math. Soc. 140 (2012), 449463
MSC (2010):
Primary 11F67, 11F12, 11F30
Published electronically:
June 14, 2011
MathSciNet review:
2846314
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Abstract: Let be a fixed selfdual HeckeMaass cusp form for and let be an orthogonal basis of holomorphic cusp forms of weight for . We prove an asymptotic formula for the first moment of the first derivative of at the central point , where runs over , , large enough. This implies that for each large enough there exists with such that .
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 [1]
 D. Bump, Automorphic Forms on , Lecture Notes in Mathematics 1083, SpringerVerlag, 1984. MR 765698 (86g:11028)
 [2]
 D. Bump, S. Friedberg and J. Hoffstein, Nonvanishing theorems for functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543618. MR 1074487 (92a:11058)
 [3]
 D. Goldfeld, Automorphic forms and functions for the group , Cambridge Studies in Advanced Mathematics 99, Cambridge Univ. Press, Cambridge, 2006. MR 2254662 (2008d:11046)
 [4]
 D. Goldfeld and X. Li, Voronoi formulas on , Int. Math. Res. Not. (2006), Art. ID 86295, 25 pages. MR 2233713 (2007f:11052)
 [5]
 A. Ivić, On the ternary additive divisor problem and the sixth moment of the zetafunction, Sieve methods, exponential sums, and their applications in number theory, London Math. Soc. Lecture Note Ser. 237, Cambridge Univ. Press, Cambridge, 1997, pp. 205243. MR 1635762 (99k:11129)
 [6]
 H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics 17, Amer. Math. Soc., Providence, RI, 1997. MR 1474964 (98e:11051)
 [7]
 H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55131. MR 1828743 (2002h:11081)
 [8]
 H. Iwaniec and P. Sarnak, The nonvanishing of central values of automorphic functions and LandauSiegel zeros, Israel J. Math. 120 (2000), 155177. MR 1815374 (2002b:11115)
 [9]
 H. Jacquet and J.A. Shalika, A nonvanishing theorem for zeta functions of , Invent. Math. 38 (1976), 116. MR 0432596 (55:5583)
 [10]
 E. Kowalski, P. Michel and J. Vanderkam, Nonvanishing of high derivatives of automorphic functions at the center of critical strip, J. Reine Angrew. Math. 526 (2000), 134. MR 1778299 (2001h:11068)
 [11]
 E. Kowalski, P. Michel and J. Vanderkam, RankinSelberg functions in the level aspect, Duke Math. J. 114 (2002), 123191. MR 1915038 (2004c:11070)
 [12]
 X. Li, The central value of the RankinSelberg functions, GAFA 18 (2009), 16601695. MR 2481739 (2010a:11087)
 [13]
 X. Li, Bounds for functions and functions, Ann. of Math. (2) 173 (2011), no. 1, 301336. MR 2753605
 [14]
 S.C. Liu, Determination of cusp forms by central values of functions, Int. Math. Res. Not. IMRN 2010, no. 21, 40254041. MR 2738349
 [15]
 W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for , Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math. 66, Part 2, Amer. Math. Soc., Providence, RI, 1999, pp. 301310. MR 1703764 (2000e:11072)
 [16]
 S.D. Miller and W. Schmid, Automorphic distributions, functions, and Voronoi summation for , Ann. of Math. (2) 164 (2006), 423488. MR 2247965 (2007j:11065)
 [17]
 M. Young, The second moment of Lfunctions, integrated, Adv. Math. 226 (2011), no. 4, 35503578. MR 2764898
 [18]
 M. Young, The second moment of Lfunctions at special points, preprint, 2009, available at arXiv:0903.1579v1.
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Additional Information
Qingfeng Sun
Affiliation:
School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, People’s Republic of China
Email:
qfsun@mail.sdu.edu.cn
DOI:
http://dx.doi.org/10.1090/S00029939201110947X
PII:
S 00029939(2011)10947X
Keywords:
Nonvanishing,
$GL(3) ×GL(2)$ $L$functions
Received by editor(s):
April 2, 2010
Received by editor(s) in revised form:
September 24, 2010, November 28, 2010, and November 29, 2010
Published electronically:
June 14, 2011
Additional Notes:
The author was supported by National Natural Science Foundation of China (grant No. 10971119).
Communicated by:
Kathrin Bringmann
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
